Communications in Mathematical Physics

, Volume 367, Issue 2, pp 517–580 | Cite as

Local Energy Weak Solutions for the Navier–Stokes Equations in the Half-Space

  • Yasunori Maekawa
  • Hideyuki Miura
  • Christophe PrangeEmail author


The purpose of this paper is to prove the existence of global in time local energy weak solutions to the Navier–Stokes equations in the half-space \({\mathbb{R}^3_+}\). Such solutions are sometimes called Lemarié–Rieusset solutions in the whole space \({\mathbb{R}^3}\). The main tool in our work is an explicit representation formula for the pressure, which is decomposed into a Helmholtz–Leray part and a harmonic part due to the boundary. We also explain how our result enables to reprove the blow-up of the scale-critical \({L^3(\mathbb{R}^3_+)}\) norm obtained by Barker and Seregin for solutions developing a singularity in finite time.


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The authors thank Tai-Peng Tsai for many helpful comments. The first author is partially supported by JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers, ’Development of Concentrated Mathematical Center Linking to Wisdom of the Next Generation’, which is organized by Mathematical Institute of Tohoku University. The second author is partially supported by JSPS Grants 25707005 and 17K05312. The third author acknowledges financial support from the French Agence Nationale de la Recherche under Grant ANR-16-CE40-0027-01, as well as from the IDEX of the University of Bordeaux for the BOLIDE project.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsKyoto UniversityKyotoJapan
  2. 2.Department of Mathematical and Computing SciencesTokyo Institute of TechnologyTokyoJapan
  3. 3.CNRS, UMR [5251], IMBUniversité de BordeauxBordeauxFrance

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